36558(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆUDC 629.5.015.5:629.544

Ivo SENJANOVIĆ

Stipe TOMAŠEVIĆ

Rajko GRUBIŠIĆ

Coupled Horizontal and Torsional Vibrations of Container Ships

Original scientifi c paper

Necessity of performing hydroelasticity analysis of large container ships is pointed out. For this purpose theory of coupled horizontal and torsional vibration is described. Beam fi nite ele-ment for vibration analysis, with eight degrees of freedom, which includes bending, shear, tor-sional and warping stiffness, is developed. Coupling between horizontal and torsional vibration is realised through inertia forces. Application of the theory is illustrated in case of a large container ship.

Keywords: container ship, horizontal and torsional vibration, fi nite element method

Spregnute horizontalne i torzijske vibracije kontejnerskih brodova

Izvorni znanstveni rad

Naglašena je potreba provođenja hidroelastične analize velikih kontejnerskih brodova. Radi toga prikazana je teorija spregnutih horizontalnih i torzijskih vibracija brodova sa širokim palub-nim otvorima. Za analizu vibracija razvijen je gredni konačni element s osam stupnjeva slobode gibanja, koji uključuje krutost na savijanje, smicanje, uvijanje i iskrivljavanje. Sprega horizontalnih i torzijskih vibracija ostvaruje se putem inercijskih sila. Primjena teorije ilustrirana je na primjeru velikoga kontejnerskog broda.

Ključne riječi: kontejnerski brod, horizontalne i torzijske vibracije, metoda konačnih elemenata

Authors’ address:

Faculty of Mechanical Engineering

and Naval Architecture

University of Zagreb

I. Lučića 5

HR-10000 Zagreb, Croatia

E-mail: ivo.senjanovic@fsb.hr

Received (Primljeno): 2007-05-31

Accepted (Prihvaćeno): 2007-07-11

Open for discussion (Otvoreno

za raspravu): 2008-12-24

1 Introduction

Increased sea transport requires building of ultra large con-tainer ships which are quite fl exible [1]. Therefore, their strength has to be checked by hydroelastic analysis [2]. Methodology of hydroelastic analysis is described in [3]. It includes defi nition of the structural model, ship and cargo mass distributions, and geometrical model of ship wetted surface. Hydroelastic analysis is based on the modal superposition method. First, dry natural vibrations of ship hull are calculated. Then modal hydrostatic stiffness, modal added mass, modal damping and modal wave load are determined. Finally, calculation of wet natural vibra-tions is performed and transfer functions for determining ship structural response to wave excitation are obtained [4], [5].

This paper deals with dry natural vibrations of container ships as an important step in their hydroelastic analysis. A ship hull, as an elastic nonprismatic thin-walled girder, performs lon-gitudinal, vertical, horizontal and torsional vibrations. Since the cross-sectional centre of gravity and deformation centre posi-tions are not identical, coupled longitudinal and vertical, as well as horizontal and torsional vibrations occur, respectively. Thus, vibration coupling is realised through the mass only.

Distance between the centre of gravity and the deformation centre is negligible for longitudinal and vertical vibration as well as for horizontal and torsional vibration of conventional ships. Therefore, in the above cases ship hull vibrations are usually analysed separately. However, the deformation centre in ships with large hatch openings is located outside the cross-section,

i.e. below the keel, and therefore the coupling of horizontal and torsional vibrations is extremely high.

The above problem is rather complicated due to geometrical discontinuity of the hull cross-section. Solution accuracy de-pends on the reliability of stiffness parameters determination, i.e. of bending, shear, torsional and warping moduli. The fi nite element method is a powerful tool to solve the above problem in a successful way.

One of the fi rst solutions for coupled horizontal and tor-sional hull vibrations, dealing with the fi nite element technique, is given in [6] and [7]. Generalised and improved solutions are presented in [8] and [9] respectively. In all these references de-termination of hull stiffness is based on the classical thin-walled girder theory, which does not give a satisfactory value for the warping modulus of the open cross-section. Besides that, the initial values of stiffness moduli are determined, so that the ap-plication of the beam theory for hull vibration analysis is limited to a few lowest natural modes.

The developed higher-order theory of thin-walled girders offers the opportunity to overcome the aforementioned defi -ciencies [10]. Within this theory the energy approach is used to defi ne hull stiffness parameters as mode dependent quantities. In this way more accurate results for the uncoupled girder vibra-tion are obtained.

In this paper the beam fi nite element method is elaborated for coupled horizontal and torsional vibration of ship hull with large hatch openings. For this purpose a beam fi nite element of eight degrees of freedom is developed. Application of the theory is illustrated in case of a large container ship.

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366 58(2007)4, 365-379

I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

2 Differential equations of beam vibrations

Referring to the fl exural beam theory [11], the total beam defl ection, w, consists of the bending defl ection, w

b, and the

shear defl ection, ws, i.e., Figure 1

. (1)

Figure 1 Beam bending and torsionSlika 1 Savijanje i uvijanje grede

The shear defl ection is a function of wb

, (2)

where E and G are the Young’s and shear modulus respective-ly, while I

b and A

s are the bending modulus and shear area re-

spectively. In (2) a term depending on mass rotation is ignored because of low infl uence and for the reason of simplicity. The angle of cross-section rotation also depends on w

b

. (3)

The cross-sectional forces include the bending moment

(4)

and the shear force

. (5)

The inertia load consists of the distributed load, qi, and the

bending moment, μi, specifi ed as

(6)

, (7)

where m is the distributed ship and added mass, Jb is the mo-

ment of inertia of ship mass about the centre line, and c is the distance between the centre of gravity and the deformation cen-tre, c = z

G – z

D, Figure 2. Deformation centre is a shear centre

i.e. torsional centre.

Figure 2 Cross section of a thin-walled girderSlika 2 Poprečni presjek tankostijenog nosača

Concerning torsion, the displacements are the twist angle ψ and its variation

(8)

as a warping measure, Figure 1. The sectional forces include the total torque, T, consisting of the pure torsional torque, T

t, and

the warping torque Tw [12] i.e.

, (9)

w w wb s= +

wEI

GA

w

xsb

s

b= − ∂∂

2

2

ϕ = ∂∂w

xb

M EIw

xbb= − ∂

∂

2

2

Q GAw

xss= ∂

∂

q mw

tc

ti = − ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟

2

2

2

2

ψ

μi bbJ

w

x t= − ∂

∂ ∂

3

2

ϑ ψ= ∂∂ x

T T Tt w= +

36758(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

where

(10)

(11)

and the bimoment given by

. (12)

The inertia load includes the distributed torque, μti, and the

bimoment, μwi

, presented in the following form:

(13)

, (14)

where Jt is polar moment of inertia of ship and added mass

about the deformation centre and Jw is bimoment of inertia of

ship mass about the warping centre, Figure 2.Considering the equilibrium of a differential element, one

may write

(15)

Substituting corresponding sectional forces and distributed loads into (15) and condensing the fi rst two equations (15), the following system of differential equations is obtained for a pris-matic beam [13], [14]:

(16)

Since

the ratio E/G may be expressed by Poisson’s ratio ν.

3 Beam fi nite element

The properties of a fi nite element for the coupled horizontal and torsional vibration analysis may be derived from the ele-ment total energy. The total energy consists of the deformation energy, the inertia energy, the work of the external lateral load, q, and the torque, μ, and the work of the boundary forces. Thus [14],

(17)

where l is the element length.Since the beam has four displacements, ω, ϕ, ψ, ϑ, a two-

nodded fi nite element has eight degrees of freedom, i.e. four nodal shear-bending and torsion-warping displacements respec-tively, Figure 3,

(18)

T GIxt t= ∂

∂ψ

T EIxw w= − ∂

∂

3

3

ψ

B EIxw w= ∂

∂

2

2

ψ

μ ψti tJ

tmc

w

t= − ∂

∂− ∂

∂

2

2

2

2

μ ψwi wJ

x t= ∂

∂ ∂

4

2 2

∂∂

= +

∂∂

= − −

∂∂

= − − −

M

xQ

Q

xq q

T

x

i

i

ti wi

μ

μ μ μ .

GE=+2 1( )ν

EIw

x

EI

GAm J

w

x t

mw

bb b

sb

b∂∂

− +⎛⎝⎜

⎞⎠⎟

∂∂ ∂

+

+ ∂

4

4

4

2 2

2bb b

s

b

w

t

J

GA

w

tc

tq

EIx

∂+ ∂

∂+ ∂

∂⎛⎝⎜

⎞⎠⎟

=

∂∂

2

4

4

2

2

4

4

ψ

ψ −− ∂∂

+ ∂∂

− ∂∂ ∂

+

+ ∂∂

GIx

Jt

Jx t

mcw

t

t t w

b

2

2

2

2

4

2 2

2

2

ψ ψ ψ

−− ∂∂ ∂

⎛⎝⎜

⎞⎠⎟

=EI

GA

w

x tb

s

b4

2 2 μ .

E

EIw

xGA

w

x

E

tot

bb

ss

=

∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

+

+

1

2

2

2

2 2

IIx

GIxw t

∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠⎟

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥

2

2

2 2ψ ψ ⎥⎥

⎥

+

+

∂∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂ ∂

⎛⎝⎜

⎞⎠⎟

+

∫ d x

mw

tJ

w

x t

l

bb

0

2 2 2

1

2++ ∂

∂∂∂

+

+ ∂∂ ∂

⎛⎝⎜

⎞⎠⎟

+ ∂∂

⎛⎝⎜

⎞⎠

2

2 2

mcw

t t

Jx t

Jtw t

ψ

ψ ψ⎟⎟

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

−

− + +

∫

∫

2

0

0

d

d

x

qw x

l

l

( )μψ (( ) ,Qw M T Bwl− + +ϕ ψ ϑ 0

U

w

w l

l

V{ } =

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

{ } =

( )

( )

( )

( )

,

( )0

0

0

ϕ

ϕ

ψϑϑψϑ

( )

( )

( )

0

l

l

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

368 58(2007)4, 365-379

I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

Figure 3 Beam fi nite elementSlika 3 Gredni konačni element

Therefore, the basic beam displacements, wb and ψ, may be

presented by the third-order polynomials

(19)

Furthermore, satisfying alternately the unit value for one of the nodal displacement {U} and zero values for the remaining displacements, and doing the same for {V}, it follows that:

(20)

where wbi, w

si, w

i and ψ

i are the shape functions given below

(21)

(22)

(23)

(24)

(25)

Substituting Eqs. (20) into (17) one obtains

(26)

where, assuming constant values of the element properties,

– bending-shear stiffness matrix,

– warping-torsion stiffness matrix,

– shear-bending mass matrix,

w ax

l

d k

b kk

kk

= { } =

= { } =

ξ ξ

ψ ξ

,

, , , , .0 1 2 3

w w U i

w w U

w w U

V

b bi

s si

i

i

= { } =

= { }= { }= { }

, , , ,

,

1 2 3 4

ψ ψ ,, . . . . . . ,= { }T

w a

w b

w c

d

bi ikk

si ikk

i ikk

i ikk

= { }= { }= { }= { }

ξ

ξ

ξ

ψ ξ

al l

ik[ ] =+

+ −− + + − +1

1 12

1 6 0 3 2

4 1 3 1 12 2 1

β

ββ β β( ) ( ) ( 33

6 0 3 2

2 1 6 0 1 6

ββ

β β β

)

( ) ( )

l l

l l l

−− − − −

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

bl l

ik[ ] =+

−+ −

−1

1 12

6 12 0 0

4 1 3 6 0 0

6 12 0 0β

β ββ β β

β β( )

22 1 6 6 0 0β β β( )− −

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥l l

dl l l

l l

ik[ ] =

−−

−−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1 0 3 2

0 2

0 0 3 2

0 0

c a bEI

GA lik ik ikb

s

[ ] = [ ] + [ ] =, β 2

EU

V

k

k

U

V

Utot

T

bs

wt

=⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

+1

2

0

01

2i

i

V

m m

m m

U

V

q

T

sb st

ts tw

⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥

⎧⎨⎩

⎫⎬⎭

−μ

⎧⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

−⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

T TU

V

P

R

U

V,

k EIw

x

w

xx GA

w

x

wbs b

bi bjl

ssi s[ ] = +∫

d

d

d

dd

d

d

d2

2

2

20

jjl

xx

dd

0∫

⎡

⎣⎢

⎤

⎦⎥

k EIx x

x GIx xwt w

i jl

ti j[ ] = +∫

d

d

d

dd

d

d

d

dd

2

2

2

20

ψ ψ ψ ψxx

l

0∫

⎡

⎣⎢

⎤

⎦⎥

m m w w x Jw

x

w

xx

sb i j

l

bbi bj

l

[ ] = +⎡

⎣⎢

⎤

⎦⎥∫ ∫d

d

d

d

dd

0 0

36958(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

– torsion-warping mass matrix,

– shear-torsion mass matrix,

(27)

The vectors {P} and {R} represent the shear-bending and torsion-warping nodal forces, respectively,

. (28)

The above matrices are specifi ed in Appendix A, as well as the load vectors for loads linearly distributed along the element, i.e.

. (29)

Also, shape functions of sectional forces are listed in Ap-pendix B.

The total element energy has to be at its minimum. Satisfy-ing the relevant conditions

(30)

and employing Lagrange equations of motion, the fi nite element equation yields

(31)

where

(32)

It is obvious that coupling between the bending and torsion occurs through the mass matrix only, by the coupling matrices [m]

st and [m]

ts.

4 Finite element transformation

In the fi nite element equation (31) fi rst the element prop-erties related to bending and then those related to torsion ap-pear. To make a standard fi nite element assembling possible, it is necessary to transform Eq. (31) in such a way that fi rst all properties related to the fi rst node and then those belonging to the second one are specifi ed. Thus, the nodal force and displace-ment vectors read

. (33)

The same transformation has to be done for the load vector {ƒ}qμ resulting in {ƒ ˜}qμ. The above vector transformation implies the row and column exchange in the stiffness and mass matrix according to the following set form:

(34)

The element defl ection refers to the deformation centre as the origin of a local coordinate system. Since the vertical posi-tion of the deformation centre varies along the ship’s hull, it is necessary to prescribe the element defl ection for a common line, in order to be able to assemble the elements. Thus, choosing the x-axis (base line) of the global coordinate system as the referent line, the following relation between the former and the latter nodal defl ections exists:

(35)

m J x Jx x

xtw t i j

l

wi j

l

[ ] = +⎡

⎣⎢

⎤

⎦⎥∫ ∫ψ ψ ψ ψ

dd

d

d

dd

0 0

m mc w x m mst i j

l

ts st

T[ ] =⎡

⎣⎢

⎤

⎦⎥ [ ] = [ ]∫ ψ d

0

,

q q w xj

l

{ } =⎧⎨⎩

⎫⎬⎭

−

{ } =

∫ d shear load vector,0

μ μ ψ jj

l

x

i j

d torsion load vector,0

1 2 3

∫⎧⎨⎩

⎫⎬⎭

−

=, , , ,, .4

P

Q

M

Q l

M l

R

T

{ } =

−

−

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

{ } =

−( )

( )

( )

( )

,

0

0

(( )

( )

( )

( )

0

0−⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

B

T l

B l

w

w

q q q= + = +0 1 0 1ξ μ μ μ ξ,

∂∂{ } = { } ∂

∂{ } = { }E

U

E

Vtot tot0 0, ,

f k m fq{ } = [ ]{ } + [ ]{ } − { }δ δ μ

fP

Rf

q U

V

k

q{ } =⎧⎨⎩

⎫⎬⎭

{ } =⎧⎨⎩

⎫⎬⎭

{ } =⎧⎨⎩

⎫⎬⎭

[ ]

, ,μ μδ

==⎡

⎣⎢

⎤

⎦⎥ [ ] =

⎡

⎣⎢

⎤

⎦⎥

k

km

m m

m mbs

wt

sb st

ts tw

0

0, .

f

Q

M

T

B

Q l

M l

T l

B l

w

w

{ } =

−

−−

−

( )

( )

( )

( )

( )

( )

( )

( )

0

0

0

0

⎧⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪

{ } =,

( )

( )

(

δ

ϕψ

w 0

0

00

0

)

( )

( )

( )

( )

( )

ϑ

ϕψϑ

w l

l

l

l

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪

⎭⎭

⎪⎪⎪⎪⎪

1 2 5 6 3 4 7 8

1 11 12 15 16 13 14 17 18

2 21 22 25 26 23 24 27 28

5 551 52 55 56 53 54 57 58

6 61 62 65 66 63 64 67 68

3 31 32 35 36 333 34 37 38

4 41 42 45 46 43 44 47 48

7 71 72 75 76 73 74 77 78

88 81 82 85 86 83 84 87 88

w w z

w l w l z lD

D

( ) ( ) ( )

( ) ( ) ( ),

0 0 0= += +

ψψ

370 58(2007)4, 365-379

I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

where zD is the coordinate of the deformation centre, Figure 2.

The other displacements are the same in both coordinate sys-tems. Twist angle ψ does not have infl uence on the cross-sec-tion rotation angle ϕ. The local displacement vector may be expressed as

(36)

where [T ˜] is the transformation matrix

(37)

Since the total element energy is not changed by the above transformations, a new element equation may be derived taking (36) into account. Thus, one obtains in the global coordinate system

, (38)

where

(39)

The fi rst of the above expressions transforms the nodal

torques into the form

(40)

5 Vibration analysis

A ship’s hull is modelled by a set of beam fi nite elements. Their assemblage in the global coordinate system, performed in the standard way, results in the matrix equation of motion, which may be extended by the damping forces

(41)

where [K], [C] and [M] are the stiffness, damping and mass matrices, respectively; {∆}, {∆̇} and {∆̈} are the displacement, velocity and acceleration vectors, respectively; and {F(t)} is the load vector.

In case of natural vibration {F(t)} = {0} and the infl uence of damping is rather low for ship structures so that the damping forces may be ignored. Assuming

(42)

where {φ} and ω are the mode vector and natural frequency respectively, Eq. (41) leads to the eigenvalue problem

(43)

which may be solved employing different numerical meth-ods [15]. The basic one is the determinant search method in which ω is found from the condition

(44)

by an iteration procedure. Afterwards, {φ} follows from (43) assuming unit value for one element in {φ}.

The forced vibration analysis may be performed by direct integration of Eq. (41) or as well as by the modal superposition method. In the latter case the displacement vector is presented in the form

, (45)

where [φ] = [{φ}] is the undamped mode matrix and {X} is the generalised displacement vector. Substituting (45) into (41), the modal equation yields

(46)

where

(47)

The matrices [k] and [m] are diagonal, while [c] becomes di-agonal only in a special case, for instance if [C] = α [M] + β [K], where α and β are coeffi cients [14].

Solving (46) for undamped natural vibration [k] = [ω2m] is obtained, and by its backward substitution into (46) the fi nal form of the modal equation yields

(48)

where

(49)

δ δ{ } = ⎡⎣ ⎤⎦ {T ]

TT

TT

zD

⎡⎣ ⎤⎦ =[ ] [ ][ ] [ ]

⎡

⎣⎢

⎤

⎦⎥ [ ] =

0

0

1 0 0

0 1 0 0

0 0 1 0,

00 0 0 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

f k m fq

{ } = ⎡⎣

⎤⎦ { } + ⎡⎣ ⎤⎦ { } − { }δ δ

μ

f T f

k T k T

T

T

{ } = ⎡⎣ ⎤⎦ { }⎡⎣

⎤⎦ = ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦

mm T m T

f T f

T

q

T

q

⎡⎣ ⎤⎦ = ⎡⎣ ⎤⎦ [ ] ⎡⎣ ⎤⎦

{ } = ⎡⎣ ⎤⎦ { }μ μμ

.

− = − −= +

T T z Q

T l T l z Q lD

D

( ) ( ) ( )

( ) ( ) ( ).

0 0 0

K C M F t[ ]{ } + [ ]{ } + [ ]{ } = { }Δ Δ Δ ( ) ,

Δ{ } = { }φ ωei t

K M[ ] − [ ]( ){ } = { }ω φ2 0

K M[ ] − [ ] =ω2 0

Δ{ } = [ ]{ }φ X

k X c X m X f t[ ]{ } + [ ]{ } + [ ]{ } = { }( )

k K

c

T[ ] = [ ] [ ][ ][ ] =

φ φ

φ

- modal stiffness matrix

[[ ] [ ][ ][ ] = [ ] [ ]

T

T

C

m M

φ

φ

- modal damping matrix

φφ

φ

[ ]{ } = [ ] { }

- modal mass matrix

- mf t F tT

( ) ( ) oodal load vector.

ω ω ζ ϕ2 2⎡⎣ ⎤⎦ { } + [ ]{ } { } + { } = { }X X X t( )

ω

ζ

[ ] =⎡

⎣⎢

⎤

⎦⎥

k

mii

ii

- natural frequency matrix

[[ ] =⎡

⎣⎢⎢

⎤

⎦⎥⎥

c

k mij

ii ii2 ( )- relative damping mmatrix

- relative loadϕ( )( )

tf t

mi

ii

{ } =⎧⎨⎩

⎫⎬⎭

vector.

37158(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

If {ζ} is diagonal, the matrix Eq. (48) is split into a set of uncoupled modal equations.

The ship vibration is caused by the engine and propeller ex-citation forces, which are of the periodical nature and therefore may be divided into harmonics. Thus, the ship’s hull response is obtained solving either (41) or (46). In both cases the system of differential equations is transformed into a system of algebraic equations.

If hull vibration is induced by waves, the time integration of (41) or (46) can be performed. A few numerical methods are available for this purpose, as for instance the Houbolt, the New-mark and the Wilson θ method [15], as well as the harmonic acceleration method [16], [17].

6 Natural vibrations of container ship

The application of developed theory is illustrated in case of a 7800 TEU VLCS (Very Large Container Ship), Figure 4. The main vessel particulars are the following:

Length over all Loa

= 334 mLength between perpendiculars L

pp = 319 m

Breadth B = 42.8 mDepth H = 24.6 mDraught T = 14.5 mDisplacement, full load ∆

f = 135336 t

Displacement, ballast ∆b = 68387 t

Engine power P = 69620 kWShip speed v = 25.4 kn

Figure 4 7800 TEU container shipSlika 4 Kontejnerski brod nosivosti 7800 TEU

Figure 5 Midship sectionSlika 5 Glavno rebro

The midship section, which shows a double skin structure with the web frames and longitudinals, is presented in Figure 5. Rows and tiers of containers at the midship section are indi-cated in Figure 6. Vertical positions of the neutral line, deforma-tion (shear, torsional) centre, and the centre of gravity are also marked in the fi gure. A large distance between the gravity centre and the deformation centre causes high coupling of horizontal and torsional vibrations.

Figure 6 Container distribution at midship sectionSlika 6 Raspored kontejnera na glavnom rebru

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I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

Figure 7 Longitudinal distribution of ship cross-section geometrical propertiesSlika 7 Uzdužna raspodjela geometrijskih značajki poprečnog presjeka brodskog trupa

37358(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

Figure 8 Longitudinal distribution of ship mass properties, full loadSlika 8 Uzdužna raspodjela masenih značajki potpuno nakrcanog broda

374 58(2007)4, 365-379

I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

bending moment My and shear force Q

z are presented in Fig-

ures 11 and 12 respectively.

Figure 10 Natural modes of vertical vibrationsSlika 10 Prirodni oblici vertikalnih vibracija

Figure 11 Vertical bending moment and shear force of the fi rst natural modeSlika 11 Vertikalni moment savijanja i poprečna sila prvog oblika vibriranja

Figure 12 Vertical bending moment and shear force of the second natural modeSlika 12 Vertikalni moment savijanja i poprečna sila drugog oblika vibriranja

Figure 9 Longitudinal distribution of ship mass properties, ballast conditionSlika 9 Uzdužna raspodjela masenih značajki broda u balastu

The ship hull stiffness properties are calculated by program STIFF [18], based on the theory of thin-walled girders [10], [19]. Their distributions along the ship are shown in Figure 7. It is evident that geometrical properties rapidly change values in the engine and superstructure area due to closed ship cross-section. This is especially pronounced in the case of torsional modulus, which takes quite small values for open cross-section and rather high for closed one, Figure 7e.

Longitudinal distributions of ship mass properties for full load and ballast conditions are shown in Figures 8 and 9 respec-tively. The harmonic variation of diagrams in Figure 8 is a result of container arrangement.

Dry natural vibrations are calculated by the modifi ed and improved program DYANA within [4]. The ship hull is divided into 50 beam fi nite elements. Finite elements of closed cross-section (6 d.o.f.) are used in the ship bow, ship aft and in the engine room area. Dry natural frequencies for vertical vibra-tions, and coupled horizontal and torsional vibrations for full load and ballast conditions are listed in Table 1. Their values for ballast condition are higher due to the lower mass. The lowest frequency value, which plays the main role in wave excitation, is detected for coupled vibrations. As a result of rather low tor-sional stiffness it belongs to primarily torsional mode.

The fi rst two dry natural modes, i.e. total defl ection w, of vertical vibrations for full load are shown in Figure 10. They are of ordinary shape. The corresponding diagrams of vertical

37558(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

Figure 13 The fi rst natural modes of horizontal and torsional vibrationsSlika 13 Prvi prirodni oblici horizontalnih i torzijskih vibracija

Figure 14 The second natural modes of horizontal and torsional vibrationsSlika 14 Drugi prirodni oblici horizontalnih i torzijskih vibracija

The fi rst two natural modes of coupled horizontal and tor-sional vibrations for full load are shown in Figures 13 and 14 re-spectively. Symbol w

G denotes horizontal hull defl ection at the

level of gravity centre, while ψ denotes torsional angle. The cor-responding sectional forces, i.e. horizontal bending moment M

z,

shear force Qy, torque T and warping bimoment B

w, are shown in

Figures 15, 16, 17 and 18 respectively. Torque T is determined with respect to the shear centre of corresponding fi nite element. T and B

w show some discontinuity in the area of the engine room

due to different type of the hull cross-sections.

Figure 15 Horizontal bending moment and shear force of the fi rst natural modeSlika 15 Horizontalni moment savijanja i poprečna sila prvog oblika vibriranja

Figure 16 Torque and warping bimoment of the fi rst natural modeSlika 16 Moment uvijanja i bimoment vitoperenja prvog prirodnog oblika

Figure 17 Horizontal bending moment and shear force of the second natural modeSlika 17 Horizontalni moment savijanja i poprečna sila drugog oblika vibriranja

Figure 18 Torque and warping bimoment of the second natural modeSlika 18 Moment uvijanja i bimoment vitoperenja drugog prirodnog oblika

Once the dry natural modes of ship hull are determined, it is possible to transfer the beam node displacements to the ship wetted surface for the hydrodynamic calculation. The transfor-

376 58(2007)4, 365-379

I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS...

mation (spreading) functions for vertical and coupled horizontal and torsional vibration yield respectively [3]

(50)

(51)

where w is hull defl ection, ψ is twist angle, ),,( ZYxuu = is the cross-section warping function reduced to the wetted surface, z

N and z

S coordinate of neutral line and shear centre

respectively, and Y and Z are coordinates of the point on the ship surface. The fi rst two dry natural modes of the ship wetted surface in case of vertical and coupled horizontal and torsional vibrations are shown in Figures 19, 20, 21 and 22.

Figure 19 The fi rst natural mode of vertical vibrations, ω

1 = 4

rad/s

Slika 19 Prvi prirodni oblik vertikalnih vibracija, ω1 = 4

rad/s

Figure 20 The second natural mode of vertical vibrations, ω

2 = 8.41

rad/s

Slika 20 Drugi prirodni oblik vertikalnih vibracija, ω2 = 8,41

rad/s

Figure 21 The fi rst natural mode of coupled horizontal and torsional vibrations, ω

1 = 2.18

rad/s

Slika 21 Prvi prirodni oblik spregnutih horizontalnih i torzijskih vibracija, ω

1 = 2,18

rad/s

Figure 22 The second natural mode of coupled horizontal and torsional vibrations, ω

2 = 4.23

rad/s

Slika 22 Drugi prirodni oblik spregnutih horizontalnih i torzijskih vibracija, ω

2 = 4,23

rad/s

Table 1 Dry natural frequencies, ωi [rad/s]

Tablica 1 Prirodne frekvencije za trup u zraku, ωi [rad/s]

Mode no.

Full load Ballast

Vertical Horizontal + torsional Vertical Horizontal +

torsional1 4.00 2.18 5.59 3.982 8.41 4.23 11.60 6.793 13.22 7.08 17.85 11.554 18.07 9.23 25.02 14.905 23.04 13.19 32.52 20.596 28.09 15.37 39.21 23.037 32.77 18.22 46.29 27.988 37.22 22.65 53.25 32.649 41.73 23.75 60.40 36.33

10 42.27 28.38 66.96 40.38

7 Conclusion

Ultra large container ships are quite fl exible and present classifi cation rules cannot be used for reliable structure design. Therefore, hydroelastic analysis has to be performed. The meth-odology of hydroelastic analysis is elaborated in [3]. One of the basic steps is dry natural vibration analysis of ship hull. Vertical vibration calculation is performed in a standard way, while the coupled horizontal and torsional vibrations are rather complex as it is evident from this article.

The theory of coupled horizontal and torsional vibrations of ships with large deck openings is checked by analytical vibra-tion solution for a prismatic pontoon [20], [21]. Furthermore, the implementation of the structural model in hydroelastic analysis is checked for the case of a fl exible barge [3], [4], for which test results are available [22]. The infl uence of transverse bulkheads

h i kvv

N v

w

xZ z w= − − +d

d( )

h i j khth

h S

w

xY

xu w Z z Y= − +

⎛⎝⎜

⎞⎠⎟

+ + −[ ] −d

d

d

d

ψ ψ ψ( )

37758(2007)4, 365-379

COUPLED HORIZONTAL AND TORSIONAL VIBRATIONS... I. SENJANOVIĆ, S. TOMAŠEVIĆ, R. GRUBIŠIĆ

of container ships on hull stiffness is investigated in [23], where an increase of torsional stiffness of 127% is determined, while the infl uence on bending stiffness is almost negligible. Reper-cussion on torsional natural frequencies is ca 6%.

The reliability of the 1D FEM model and the software code is confi rmed by 3D FEM analysis of dry vibrations for ship lightweight with displacement of 33 692 t. Natural frequencies of the basic modes are compared in Table 2 [5].

Table 2 Natural frequencies of dry vibrations, ωi [rad/s] (T – tor-

sion, HB – horizontal bending)Tablica 2 Prirodne frekvencije za trup u zraku, ω

i [rad/s] (T – uvi-

janje, HB – horizontalno savijanje)

Mode No. 1 (T) 2 (T+HB)1D FEM model 5.39 9.233D FEM model 5.41 9.42

The performed vibration analysis of the 7800 TEU contain-er ship shows that the developed theory, utilising the 1D FEM model and the theory of thin-walled girders for determination of hull stiffness parameters, is an effi cient tool for application in ship hydroelastic analyses.

Acknowledgment

The authors would like to express their gratitude to Ms. Es-telle Stumpf, research engineer at Bureau Veritas, Marine Divi-sion - Research Department, for performing the 3D FEM vibra-tion analysis of container ship and king permission to publish the valuable results.

References

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[9] PEDERSEN, P.T.: “Torsional response of container ships” Journal of Ship Research, Vol. 29, 1985, p. 194-205.

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[11] SENJANOVIĆ, I., FAN, Y.: “A higher-order fl exural beam theory”, Computers & Structures, 32(1989) 5, p. 973-986.

[12] SENJANOVIĆ, I., FAN, Y.: “A higher-order torsional beam theory”, Engineering Modelling, 10(1997) 1-4, p. 25-40.

[13] TIMOSHENKO, S., YOUNG, D. H.: “Vibration Problems in Engineering”, D. Van Nostrand, 1955.

[14] SENJANOVIĆ, I.: “Ship Vibration”, 2nd Part., University of Zagreb, Zagreb, (in Croatian), 1990.

[15] BATHE, K.J.: “Finite Element Procedures”, Prentice Hall, 1996.

[16] SENJANOVIĆ, I.: “Harmonic acceleration method for dynamic structural analysis”, Computers & Structures, 18(1984) 1, p. 71-80.

[17] LOZINA, Ž.: “A comparison of harmonic acceleration method with the other commonly used methods for calcu-lation of dynamic transient response”, Computers & Struc-tures, 29(1988) 2, p. 227-240.

[18] FAN, Y., SENJANOVIĆ, I.: “STIFF User’s manual”, FAMENA, Zagreb, 1990.

[19] SENJANOVIĆ, I., FAN, Y.: “A fi nite element formula-tion of initial ship cross-section properties”, Brodogradnja 41(1993) 1, p. 27-36.

[20] SENJANOVIĆ, I., ĆATIPOVIĆ, I., TOMAŠEVIĆ, S.: “Coupled horizontal and torsional vibrations of a fl exible barge, Engineering Structures 30(2008) 93-109.

[21] SENJANOVIĆ, I., ĆATIPOVIĆ, I., TOMAŠEVIĆ, S.: “Coupled fl exural and torsional vibrations of ship-like girders, 45(2007) 1002-1021.

[22] REMY, F., MOLIN, B., LEDOUX, A.: “Experimental and numerical study of the wave response of a fl exible barge”, Hydroelasticity in Marine Technology, Wuxi, China, 2006., p. 255-264.

[23] SENJANOVIĆ, I., SENJANOVIĆ, T., TOMAŠEVIĆ, S., RUDAN, S: “Contribution of transverse bulkheads to hull stiffness of large container ships”, Brodogradnja (submit-ted).